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Universal continuous transition to turbulence in a planar shear flow

Published online by Cambridge University Press:  04 July 2017

Matthew Chantry ,
Laurette S. Tuckerman Open the ORCID record for Laurette S. Tuckerman [Opens in a new window]  and
Dwight Barkley
Matthew Chantry
Affiliation:
Atmospheric, Oceanic and Planetary Physics, University of Oxford, Clarendon Laboratory, Parks Road, Oxford OX1 3PU, UK Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, France Kavli Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Laurette S. Tuckerman*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS, ESPCI Paris, PSL Research University, Sorbonne Université, Univ. Paris Diderot, France Kavli Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
Dwight Barkley
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Kavli Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email address for correspondence: laurette@pmmh.espci.fr
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Abstract

We examine the onset of turbulence in Waleffe flow – the planar shear flow between stress-free boundaries driven by a sinusoidal body force. By truncating the wall-normal representation to four modes, we are able to simulate system sizes an order of magnitude larger than any previously simulated, and thereby to attack the question of universality for a planar shear flow. We demonstrate that the equilibrium turbulence fraction increases continuously from zero above a critical Reynolds number and that statistics of the turbulent structures exhibit the power-law scalings of the (2 + 1)-D directed-percolation universality class.

JFM classification

Instability: Instability Instability: Transition to turbulence

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Rapids
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Journal of Fluid Mechanics , Volume 824 , 10 August 2017 , R1
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Copyright
© 2017 Cambridge University Press 

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Chantry et al. supplementary movie

Evolution of turbulence fraction in a domain of streamwise and spanwise size $2560h imes 2560h$. Left (black box): $Re=173.808$, very near $Re_c=173.80$. Right (red box): $Re=173.888$, above $Re_c$. Middle: Power law $F \sim t^{-0.4505}$ (blue line). For $Re\approx Re_c$ (black curve), the power law is followed throughout the evolution shown. For $Re>Re_c$ (red curve), after an initial power-law decay, the turbulence fraction saturates at a finite value.

Download Chantry et al. supplementary movie(Video)
Video 265.7 MB